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algebra

mathematics, logic 1. A loose term for an algebraicstructure.

2. A vector space that is also a ring, where the vector

space and the ring share the same addition operation and are

related in certain other ways.

An example algebra is the set of 2x2 matrices with realnumbers as entries, with the usual operations of addition and

matrix multiplication, and the usual scalar multiplication.

Another example is the set of all polynomials with real

coefficients, with the usual operations.

In more detail, we have:

(1) an underlying set,

(3) an operation of scalar multiplication, whose input is a

scalar and a member of the underlying set and whose output is

a member of the underlying set, just as in a vector space,

(4) an operation of addition of members of the underlying set,

whose input is an ordered pair of such members and whose

output is one such member, just as in a vector space or a

ring,

(5) an operation of multiplication of members of the

underlying set, whose input is an ordered pair of such members

and whose output is one such member, just as in a ring.

This whole thing constitutes an `algebra' iff:

(1) it is a vector space if you discard item (5) and

(2) it is a ring if you discard (2) and (3) and

(3) for any scalar r and any two members A, B of the

underlying set we have r(AB) = (rA)B = A(rB). In other words

it doesn't matter whether you multiply members of the algebra

first and then multiply by the scalar, or multiply one of them

by the scalar first and then multiply the two members of the

algebra. Note that the A comes before the B because the

multiplication is in some cases not commutative, e.g. the

matrix example.

Another example (an example of a Banach algebra) is the set

the usual norm. The multiplication is the operation of

composition of operators, and the addition and scalar

multiplication are just what you would expect.

Two other examples are tensor algebras and Cliffordalgebras.